## 1. Introduction

[2] Analysis of radio wave scattering from planetary surfaces allows the inference of statistical surface properties on a range of relevant structural scales that are conditioned by the wavelength of observations. Scattering laws describe the variation of the radar cross section (RCS) with the angles of incidence and scattering in the bistatic case, and with the angle of incidence in the monostatic, or backscatter, case. Since scattering laws are typically parameterized by physical surface properties, such as the Fresnel reflection coefficient, the surface root-mean-square (RMS) height, and the surface RMS slope, they provide a means for estimating such surface parameters by comparison of theory with observations. Knowledge of these parameters can be used to constrain the composition, structure, and, in some instances, the natural history of the surface.

[3] The eponymic backscattering law developed by *Hagfors* [1964], which we denote σ°_{H}(θ), has become widely used in planetary radar studies because it is, in many instances, in close agreement with data [e.g., *Simpson and Tyler*, 1982]. The width parameter “*C*” of σ°_{H}(θ) provides a measure of the angular extent of the backscattering lobe and hence the roughness of the surface. In addition, the backscatter strength at normal incidence to the mean surface is given by ρ*C*/2, where ρ is the power reflection coefficient. In this work we discuss some theoretical aspects of σ°_{H}(θ) pertinent to the analysis of remote sensing data.

[4] Notwithstanding its success in matching observational data, the use of σ°_{H}(θ) has been subject to several objections. The most notable of these, that use of the exponential correlation function is entirely inappropriate on both mathematical and physical grounds, was raised by *Barrick* [1970] whose critique is addressed in several places in this paper. Although the central issues have remained unresolved in the literature, the controversy over the adequacy of σ°_{H}(θ) has subsided. It can be said that σ°_{H}(θ) is now the de facto scattering law employed in most surface studies of the terrestrial planets, and that Hagfors' parameter *C* is the standard measure of surface roughness applied to the terrestrial planets and other similarly scattering bodies.

[5] The derivation of σ°_{H}(θ) assumes a classically modeled surface of homogeneous material with Gaussian height distribution and an exponential correlation function. The Kirchhoff approximation (KA) is used to estimate the fields induced on the surface by the incident electromagnetic radiation. Hence Hagfors' result is restricted by the limitations of KA, among other things. This is manifest, for example, in the failure of σ°_{H}(θ) to conserve energy because KA cannot capture the full range of surface scattering behaviors, as discussed below.

[6] In analyses involving the use of σ°_{H}(θ) to represent the variation of backscatter RCS with angle, the inverse of the square root of the width parameter *C* of σ°_{H}(θ) is usually taken as the RMS slope of the surface, following a derivation by *Hagfors* [1970]. We discuss the basis for this result by investigating the scattering integral leading to σ°_{H}(θ). We show how changing the underlying assumption in a reasonable way yields different relations between the width parameter *C* and the surface RMS slope, thereby illustrating an indefiniteness inherent in the value of the surface RMS slope inferred on the basis of σ°_{H}(θ).

[7] Fractal geometry has been shown to provide more accurate representations of natural surfaces than do standard stochastic models [e.g., *Mandelbrot*, 1982; *Voss*, 1985; *Falconer*, 1990; *Barnsley*, 2000]. Consequently, fractal geometry has become increasingly attractive both as a model for physical surfaces and as the basis for a new generation of radio wave scattering models. In particular, the use of the fractional Brownian (FB) fractal surface model and the Kirchhoff approximation gives a scattering law with the same functional form as σ°_{H}(θ) when the Hurst exponent characterizing the model equals 1/2. Fractal-based scattering laws parameterized by the Hurst exponent yield a different result for the surface RMS slope which appears more physically sound than does the *C*^{−1/2} result now in common use. In contrast with the classical result, the fractal-based model leads to an RMS slope estimate explicitly linked to horizontal surface scales. This latter result is useful because the explicit variation of surface parameters with the scale at which they are measured or estimated is a fundamental characteristic of natural surfaces. Furthermore, it leads to scale-dependent scattering properties that correspond to certain aspects of planetary radar observations. Fractal-based laws, however, also fail to conserve energy as a result of employing KA to represent the boundary conditions.

[8] The remainder of this paper is organized as follows. In section 2, we provide a general discussion of scattering laws and their derivation; the Kirchhoff approximation and its reduction to σ°_{H}(θ) are summarized. Section 3 is devoted to an analysis of the integral leading to σ°_{H}(θ). We emphasize the surface scales that contribute most strongly to the value of the scattering integral and hence comprise the physical range of surface scales that dominate the scattering process, granted the validity of the several assumptions involved in the derivation of σ°_{H}(θ). In section 4, the conservation of energy aspects of σ°_{H}(θ) are studied by recourse to the bistatic version. We show that the law does not conserve energy following an analysis provided by *Barrick* [1970]. In section 5, we compare σ°_{H}(θ) with scattering laws derived on the basis of the FB surface models emphasizing the difference in the treatment of surface scales in the two approaches. As with σ°_{H}(θ), the fractal-based laws require a minimum illuminated surface size in order that the estimation error in the value of the surface RCS, resulting from a mathematical approximation, not exceed a specified value. A numerical example pertinent to the Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS) illustrates this requirement. Section 6 contains our conclusions.